Irish Math. Soc. Bulletin 56 (2005), 21–28
21
Search Techniques and Epimorphisms
Between Certain Groups and Fibonacci Groups
C. M. CAMPBELL AND P. P. CAMPBELL
Abstract. We examine the Fibonacci lengths of all generating pairs for certain centro-polyhedral groups. The problem
requires a variety of approaches both exhausive and random
search.
1. Introduction
The Fibonacci group F (r, n) is the group defined by the presentation
h a1 , a2 , . . . , an | a1 a2 · · · ar = ar+1 ,
a2 a3 · · · ar+1 = ar+2 ,
......,
an−1 an a1 · · · ar−2 = ar−1 ,
an a1 a2 · · · ar−1 = ar i
where r > 0, n > 0 and all subscripts are assumed to be reduced
modulo n. For a survey on Fibonacci groups see [13]. It is known
that all finite and some infinite groups are homomorphic images of
some Fibonacci groups, see [10] and [2]. In order to find which
Fibonacci groups occur, the concept of Fibonacci length is used.
Let G be a finitely generated group, G = hAi, where A = (a1 , . . . , an )
an ordered n-tuple. Then we have:
Definition 1. The Fibonacci orbit of G with respect to the generating n-tuple
QnA, written FA (G), is the sequence x1 = a1 , . . . , xn =
an , xi+n = j=1 xi+j−1 , i ≥ 1.
2000 Mathematics Subject Classification. 20F05.
Key words and phrases. Group, Fibonacci length, Fibonacci sequence,
centro-polyhedral group.
22
C. M. Campbell and P. P. Campbell
Definition 2. If FA (G) is periodic then the length of the period of
the sequence is called the Fibonacci length of G with respect to the
generating n-tuple A, written LENA (G). If FA (G) is not periodic
then we say that the group G has infinite Fibonacci length on the
generating n-tuple A, written LENA (G) = ∞.
We will write LEN (G) when it is clear which generating n-tuple
is being used. It is also important to note that the Fibonacci length
of a group depends on the chosen generating n-tuple.
From the theory of group presentations and, in particular, von
Dyck’s Theorem, see [12], it is possible to prove the following theorem:
Theorem 1. Let G be a group with generating n-tuple (a1 , a2 , . . . , an )
and let LENA (G) = m for finite m. Then G is an epimorphic image
of the Fibonacci group F (n, m).
For example, a 2-generator presentation for the quaternion group
Q8 is given by
h a, b | aba = b, bab = a i
and LEN{a,b} Q8 = 3. Hence Q8 is an epimorphic image of the
Fibonacci group F (2, 3) which is again Q8 . However a 3-generator
presentation for Q8 is
h a, b, c | ab = c, bc = a, ca = b i
and LEN{a,b,c} Q8 = 8. Hence Q8 is an epimomorphic image of
the infinite Fibonacci group F (3, 8) which, of course, has the finite
Fibonacci group F (3, 2) ( ∼
= F (2, 3) ) as an epimorphic image. Thus
given a group, G say, we can, where appropriate, find a Fibonacci
group of which G is an image. We also use von Dyck’s Theorem to
prove:
Theorem 2. Let G be a group defined by the presentation h X | R i.
If LENX (G) = n and H is a factor group of G on the same set of
generating symbols, then LENX (H) | LENX (G).
In order to compare methods we will use polyhedral groups and
the related centro-polyhedral groups. These families of groups have
well known connections and properties, see [6]. Polyhedral groups
and centro-polyhedral groups are defined as follows:
Fibonacci Groups
23
Definition 3. The polyhedral group (`, m, n), for `, m, n ∈ Z is defined by the presentation
h x, y, z | x` = y m = z n = xyz = 1 i.
Definition 4. The centro-polyhedral group h`, m, ni, for `, m, n ∈ Z
is defined by the presentation
h x, y, z | x` = y m = z n = xyz i.
In this paper we report on our experiments designed to calculate
all Fibonacci lengths on all n-tuples when n = 2 of certain finite
polyhedral and centro-polyhedral groups. The task of calculating
all Fibonacci lengths over all generating n-tuples when n = 2 and
when n = 3 of a non-abelian group has only been attempted, to
the authors knowledge, in [4] where the Fibonacci lengths of D2n ,
the dihedral groups of order 2n, and Q2n were calculated due to a
nice property of their automorphism groups and in [3] where certain
centro-polyhedral groups were considered. For related results see also
[1] where Fibonacci lengths for p-groups are considered. The present
paper completes the work of [3]. It started out as a straightforward
generalisation of that paper but immediately ran into problems of
computing time and resources requiring the various techniques described in the next section. The final section of the paper raises some
open problems.
All computer calculations were carried out using a standard download of the computational algebra system GAP, see [8], together with
the coset enumeration package ACE, see [7].
2. Methods
In order to calculate all Fibonacci lengths of a given group on all
generating pairs the authors wrote several programs and compared
the results and efficiency of each program against the others. One of
the main problems that occurred was that there is no known result
that will predict, with any great accuracy, the number of distinct
Fibonacci lengths that a given group might have. Our programs are
denoted by the names full exhausive search, restrictive full exhausive
search, restrictive search and random search. A brief description
of each method together with its advantages and disadvantages is
presented below.
24
C. M. Campbell and P. P. Campbell
(1) Full Exhausive Search: Here we simply choose every pair
of elements except for the obvious exceptions. If they generate the group then we calculate the Fibonacci length. This
calculation is certain to complete given that the given group
is finite but the time to complete can be prohibitively long.
(2) Restricted Full Exhausive Search: Here we first calculate the Euler function of the given group φ2 (G), see [9],
and then test all generating pairs, stopping when we have
calculated φ2 (G) distinct pairs. Thus we do not have to calculate all possible Fibonacci lengths; see [4]. This can speed
up the running of the program when compared to the Full
Exhausive Search method. Unfortunately calculating φ2 (G)
is non-trivial and can take longer to calculate than the Full
Exhausive Search method takes to complete.
(3) Restrictive Search: Using the results given in [4] it is
possible to know from the computing results how many Fibonacci lengths there are. Unfortunately this method requires the calculation of the automorphism group of the
group under investigation which is known to be hard in
general. In our experiments this method proved to be the
longest to complete and was not used much.
(4) Random Search: Here we search over a known number
of randomly chosen generating pairs and calculate the Fibonacci lengths. This proved to be very fast to compute.
The main problem was that nothing is known about the distribution of Fibonacci lengths within a group.
In all the above methods the groups, given by finite presentations,
were first converted into the isomorphic permutation groups.
3. Results
In deciding which groups to examine we chose to use the centropolyhedral groups of the form h±2, Xi or (±2, X), where
X = {±3, ±3}, {±3, ±4}
or X = {±3, ±5}.
Using the Full Exhausive Search method we were able to find all
Fibonacci lengths for the following groups and the distribution of
the Fibonacci lengths within each group.
Fibonacci Groups
25
P
|hPi|
LEN(a,b) (hPi)
Centro-polyhedral groups
hx, y, z|x2 = y 3 = z 3 = xyzi
hx, y, z|x−2 = y 3 = z 3 = xyzi
24
120
hx, y, z|x2 = y −3 = z 3 = xyzi
hx, y, z|x−2 = y −3 = z 3 = xyzi
hx, y, z|x2 = y −3 = z −3 = xyzi
hx, y, z|x−2 = y −3 = z −3 = xyzi
72
216
168
312
16 (96), 48 (288)
16 (384), 48 (1152),
80 (1920), 240 (5760)
48 (3456)
144 (31104)
16 (4608), 48 (13824)
112 (16128), 336 (48384)
The polyhedral group (2, 3, 3)
hx, y, z|x2 = y 3 = z 3 = xyz = 1i
12
16 (96)
P
|hPi|
LEN(a,b) (hPi)
Centro-polyhedral groups
hx, y, z|x2 = y 3 = z 4 = xyzi
hx, y, z|x−2 = y 3 = z 4 = xyzi
hx, y, z|x2 = y −3 = z 4 = xyzi
hx, y, z|x−2 = y −3 = z 4 = xyzi
hx, y, z|x2 = y 3 = z −4 = xyzi
hx, y, z|x−2 = y 3 = z −4 = xyzi
hx, y, z|x2 = y −3 = z −4 = xyzi
hx, y, z|x−2 = y −3 = z −4 = xyzi
48
528
336
912
240
816
624
1200
18 (864)
90 (103680)
144 (41472)
18 (311040)
36 (3456), 108 (17280)
36 (248832)
252 (145152)
180 (86400), 900 (432000)
The polyhedral group (2, 3, 4)
hx, y, z|x2 = y 3 = z 4 = xyz = 1i
24
18 (216)
P
Centro-polyhedral groups
hx, y, z|x2 = y 3 = z 5 = xyzi
|hPi|
LEN(a,b) (hPi)
120
hx, y, z|x2 = y −3 = z 5 = xyzi
hx, y, z|x2 = y −3 = z −5 = xyzi
2280
3720
12 (960), 14 (840), 42 (2520),
50 (1200), 150 (3600)
50 (144000), 60 (115200), 70 (100800),
150 (432000), 210 (302400)
36 (345600), 126 (1209600), 450 (1728000)
60 (921600), 150 (4608000), 210 (3225600)
The polyhedral group (2, 3, 5)
hx, y, z|x2 = y 3 = z 5 = xyz = 1i
60
12 (240), 14 (840), 50 (1200)
2
3
hx, y, z|x = y = z
−5
= xyzi
1320
The number in brackets indicates the total number of distinct
pairs with the stated Fibonacci length.
It is interesting to note that the above groups proved particularly
amenable to the random search method. On each run of the random
26
C. M. Campbell and P. P. Campbell
search method a complete list of Fibonacci lengths was calculated,
while the full exhausive search method took many times longer to
obtain the same results. It is also interesting to note that in Section 3.3 of [4] the random method was able, on one run, to find all
Fibonacci lengths on generating pairs.
In this section we have used several of the methods to obtain the
results in the tables. For certain of the more manageable groups we
have used both the Full Exhausive Search method and the Random
Search method in order to illustrate the power of the Random Search
method. For example, to obtain the complete results of the first table above (where we considered the groups of the form h±2, ±3, ±3i )
the Full Exhausive Search method required several days whereas the
Random Search method was completed in tens of minutes. In practice the Restrictive Full Exhausive Search method was not used as
the computation of φ2 (G) was too demanding on computer memory.
Likewise the Restrictive Search method was not used as Aut(G) was
too hard to compute in reasonable time. For the other two methods,
the Random Search method and Full Exhausive method were used
for all groups.
4. Further Work
The area of calculating Fibonacci lengths is not very well developed
and requires more attention. Of particular interest one would like to
know the following:
(1) Is it possible to predict the distribution of Fibonacci lengths
within a particular group? Why is the random search method
so efficient in obtaining the complete list of Fibonacci lengths?
(2) What general theories can be obtained regarding the Fibonacci lengths of a general group? For example does there
exist a decision process to determine whether, or not, a given
group has finite Fibonacci length?
(3) What generalisations of Fibonacci length are possible? It
seems likely that a theory may exist for groups defined by a
presentation whose relators are given via a recurrence relation.
(4) Resolve Wall’s conjecture, see [14] and [5]. The conjecture
is that k(p2 ) = pk(p), where p is an odd prime. (This holds
in all known cases. The other possibility would be k(p2 ) =
k(p).)
Fibonacci Groups
27
(5) The problem of finding all the Fibonacci lengths on n-tuples
of a given group could be parallelised. In order to find the
Fibonacci lengths of infinite groups it would be useful to
have a program that would use the Knuth-Bendix method
to find Fibonacci lengths, see [11].
References
[1] H. Aydin and G. C. Smith, Finite p-quotients of some cyclically presented
groups, J. London Math. Soc. 49 (1994), 83–92.
[2] C. M. Campbell, P. P. Campbell, H. Doostie and E. F. Robertson, “On
the Fibonacci length of powers of dihedral groups”, in: Applications of
Fibonacci numbers, Vol. 9 (edited by F. T. Howard), Kluwer, Dordrecht
(2004), 69–85.
[3] C. M. Campbell and P. P. Campbell, “The Fibonacci length of certain
centro-polyhedral groups”, J. Appl. Math. and Computing, to appear.
[4] C. M. Campbell, H. Doostie and E. F. Robertson, “Fibonacci length of
generating pairs in groups”, in: Applications of Fibonacci numbers, Vol.
3, (edited by G.A. Bergum, A. N. Phiippou and A. F. Horadam), Kluwer,
Dordrecht (1990), 27–35.
[5] P. P. Campbell, Wall numbers for the first 100,001 prime numbers, CIRCA
Technical Report University of St Andrews 2004/10 (2004).
[6] J.H. Conway, H. S. M. Coxeter and G. C. Shephard, “The centre of a finitely
generated group”, Tensor (N.S.) 25 (1972), 405–418; erratum, ibid. (N.S.)
26 (1972), 477.
[7] G. Gamble, A. Hulpke, G. Havas and C. Ramsay, ACE - Advanced Coset
Enumeration, Version 4.1.
(http://www.math.rwth-aachen.de/~Greg.Gamble/ACE)
[8] The GAP Group, GAP - Groups, Algorithms and Programming, Version
4.3 Aachen, St Andrews, 2002. (http://www.gap-system.org)
[9] P. Hall, “The Eulerian function of a group”, Quart. J. Math 7 (1936), 134–
151.
[10] D. L. Johnson, J. W. Wamsley and D. Wright, “The Fibonacci groups”,
Proc. London Math. Soc. 29 (1974), 577–592.
[11] D. E. Knuth and P. B. Bendix, “Simple word problems in universal algebra”, in Computational problems in abstract algebra, (edited by J. Leech),
Pergamon Press, Oxford (1970), 263–297.
[12] C. C. Sims, Computation with finitely presented groups, Encyclopedia of
mathematics and its applications 48, Cambridge University Press, Cambridge, 1994.
[13] R. M. Thomas, “The Fibonacci groups revisited”, in Groups St Andrews
1989 Volume 2, (edited by C. M. Campbell and E. F. Robertson), Cambridge University Press, Cambridge (1991), 445–454.
[14] D. D. Wall, “Fibonacci series modulo m”, Amer. Math. Monthly 67 (1960),
525–532.
28
C. M. Campbell and P. P. Campbell
C. M. Campbell and P. P. Campbell,
Mathematical Institute,
University of St. Andrews,
North Haugh,
St. Andrews KY16 9SS,
Scotland
colinc@mcs.st-andrews.ac.uk
peterc@mcs.st-andrews.ac.uk